isTotallyDisconnected — Mathlib

English

A countable subset of a metric space is totally disconnected.

Русский

Счётное подмножество метрического пространства totally disconnected.

LaTeX

$$$s \\subset X$ with $s$ countable and $X$ metric implies $s$ is TotallyDisconnected$$

Lean4

/-- Countable subsets of metric spaces are totally disconnected. -/
theorem isTotallyDisconnected [MetricSpace X] {s : Set X} (hs : s.Countable) : IsTotallyDisconnected s :=
  by
  rw [← totallyDisconnectedSpace_subtype_iff]
  have : Countable s := hs
  infer_instance

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